Question: You have found the following ages (in years) of all 5 sloths at your local zoo: $ 15,\enspace 10,\enspace 4,\enspace 4,\enspace 24$ What is the average age of the sloths at your zoo? What is the variance? You may round your answers to the nearest tenth.
Solution: Because we have data for all 5 sloths at the zoo, we are able to calculate the population mean $({\mu})$ and population variance $({\sigma^2})$ To find the population mean , add up the values of all $5$ ages and divide by $5$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5}} $ $ {\mu} = \dfrac{15 + 10 + 4 + 4 + 24}{{5}} = {11.4\text{ years old}} $ Find the squared deviations from the mean for each sloth. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $15$ years $3.6$ years $12.96$ years $^2$ $10$ years $-1.4$ years $1.96$ years $^2$ $4$ years $-7.4$ years $54.76$ years $^2$ $4$ years $-7.4$ years $54.76$ years $^2$ $24$ years $12.6$ years $158.76$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{12.96} + {1.96} + {54.76} + {54.76} + {158.76}} {{5}} $ $ {\sigma^2} = \dfrac{{283.2}}{{5}} = {56.64\text{ years}^2} $ The average sloth at the zoo is 11.4 years old. The population variance is 56.64 years $^2$.